0,2

The members are all palindromic in binary, i.e. a subset of A006995. - R. Stephan, Sep 28 2004

a(2n+1) = 3 * a(2n), as follows from a(n)=product_{k in K} (1+2^(2^k)), where K is the set of integers such that n=sum_{k in K} 2^k. -Emmanuel Ferrand, Sep 28 2004

J. H. Conway writes (in Math Forum): at least the first 31 numbers give odd-sided constructible polygons. See also A047999. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 19 2005

Decimal number generated by the binary bits of the n-th generation of the rule 60 elementary cellular automaton. Thus: 1; 0, 1, 1; 0, 0, 1, 0, 1; 0, 0, 0, 1, 1, 1, 1; 0, 0, 0, 0, 1, 0, 0, 0, 1; .. - Eric Weisstein (eric(AT)weisstein.com), Apr 08, 2006

One can generate this sequence using simple bitwise operations: a(n) = n XOR ( n << 1 ) where XOR is bitwise XOR and << is bitwise shift left - Joel Madigan (dochoncho(AT)gmail.com), Dec 03 2007

limit(n->inf) log(a(n)-1)/log(3) = n*log(2)/log(3) - Bret Mulvey (bret_x(AT)hotmail.com), May 17 2008

H. W. Gould, Exponential Binomial Coefficient Series. Tech. Rep. 4, Math. Dept., West Virginia Univ., Morgantown, WV, Sept. 1961.

R. K. Guy, The second strong law of small numbers. Math. Mag. 63 (1990), no. 1, 3-20.

D. Hewgill, A relationship between Pascal's triangle and Fermat numbers, Fib. Quart., 15 (1977), 183-184.

J.-P. Allouche & J. Shallit, Automatic sequences, Cambridge Univeristy Press, 2003, p 113

T. D. Noe, Table of n, a(n) for n=0..300

Index entries for sequences related to cellular automata

Dr. Math, Regular polygon formulas

Eric Weisstein's World of Mathematics, Rule 60

Eric Weisstein's World of Mathematics, Rule 102

a(n+1) = a(n) XOR 2a(n), where XOR is binary exclusive OR operator. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 27 2003

a(n)=prod(e(j, n)=1, 2^(2^j)+1) where e(j, n) is the j-th least significatif digit in binary representation of n (Roberts : see Allouche & Shallit) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 08 2004

a(2n+1) = 3a(2n). Proof: Since a(n)=product_{k in K} (1+2^(2^k)), where K is the set of integers such that n=sum_{k in K} 2^k, clearly K(2n+1) = K(2n) union {0}, hence a(2n+1)=(1+2^(2^0))*a(2n)=3*a(2n). - Emmanuel Ferrand, Sep 28 2004 - R. Stephan, Sep 28 2004

a(32*n) = 3 ^ (32 * n * log(2) / log(3)) + 1 - Bret Mulvey (bret_x(AT)hotmail.com), May 17 2008

Given a(5)=51, a(6)=85 since a(5) XOR 2a(5) = 51 XOR 102 = 85.

A001317 := proc(n) local k; add((binomial(n, k) mod 2)*2^k, k=0..n); end;

a[n_] := BitXor[ n, BitShiftLeft[ n, 1]] Table[ Nest[a, 1, x], {x, 0, 12} ] - Joel Madigan (dochoncho(AT)gmail.com), Dec 03 2007

(PARI) a(n)=sum(i=0, n, (binomial(n, i)%2)*2^i)

Cf. A000215 (Fermat numbers). Odd-numbered terms give A038183 (1D Cellular Automata rule 90, "sigma minus")

Not the same as A053576 nor as A045544.

Cf. A047999, A054432.

Sequence in context: A003527 A004729 A045544 this_sequence A053576 A077406 A054432

Adjacent sequences: A001314 A001315 A001316 this_sequence A001318 A001319 A001320

nonn,base,easy,nice

njas